Week 12 Study Guide — Shear Stress & Strain + Stress-Strain Curve

Directly supported by notes

These topics are explicitly covered in Lecture12_CTP1.pdf (24 slides):

Topic	Slide(s)	Source coverage
Yield / UTS / fracture / proportional & elastic limits	3	Definitions; “essentially coincide at yield” note
Strain energy, toughness, modulus of resilience	4-5	Integral formulas, material-class comparison
Strain hardening	6-7	Unload parallel to elastic line; $E$ unchanged; worked example (450 → 600 MPa)
Direct shear ${\tau }_{avg}=V/A$	9	Definition and orientation rule
Glued double-shear joint	10	Worked example, $F=20$ kN, $A=1000$ mm²
Single vs double shear bookkeeping	11-12	$V=F$ vs $V=F/2$
Sizing simple connections	13-14	${\tau }_{allow}={\tau }_{fail}/SF$; embedded rod $\ell =P/({\tau }_{allow}\pi d)$
Disk through a hole	15	Worked example, $P=20$ kN, $d=40$ mm, ${\tau }_{allow}=35$ MPa
Shear strain ${\gamma }_{nt}=\pi /2-{\theta }^{\prime }$	16	Definition, limit form, sign convention
Plate deformation example	17	300×400 mm with 3 mm + 2 mm displacements
Small-strain approximations	18	$\sin \gamma \approx \tan \gamma \approx \gamma$, $\cos \gamma \approx 1$
Parallelogram shear-strain example	19	400×300 mm with 5 mm + 5 mm displacements
Shear modulus + $E$-$G$-$\nu$ relation	20	$G=\tau /\gamma$, $G=E/[2(1+\nu )]$, material table
Polymer block worked example	21	$\delta =2$ mm, $P=2$ kN

The lecture expects you to be able to:

1. Identify points on a stress-strain curve and compute toughness / modulus of resilience as areas.
2. Distinguish single vs double shear from a free-body diagram and apply ${\tau }_{avg}=V/A$.
3. Size a bolted, pinned, glued, or embedded connection using ${\tau }_{allow}={\tau }_{fail}/SF$.
4. Compute shear strain at a point using small-angle approximations.
5. Use $G=\tau /\gamma$ and $G=E/[2(1+\nu )]$ to move between $E$, $G$ and $\nu$.

Strongly inferred from lecture materials

The slides imply but don’t fully derive:

• The geometric argument for why $G=E/[2(1+\nu )]$ — likely stated as a result, not proven.
• Sign convention for $\gamma$: negative when ${\theta }^{\prime }>\pi /2$ (the right angle opens up). Slide 16 mentions the sign convention; details are left for the workshop.
• The Mastering Physics homework on “Shear Stress and Strain” and “Material Properties” referenced on slide 23 — confirms the expected level of computation.

Possible lecture content (not in notes)

The slides do not explicitly cover but the topic may extend to:

• Pure shear states and Mohr’s-circle representation (likely future-year material).
• Combined axial + shear loading and the resulting principal-stress analysis.
• Detailed treatment of strain hardening models (Ludwik, Hollomon) in materials engineering follow-on courses.

Gaps requiring official source check

• Slide 17 (Example 4) is ambiguous about whether the 2 mm shortening contributes to shear strain at $A$ or only to a normal strain elsewhere — both interpretations are written up in the lecture reconstruction, but worth confirming against the original drawing.
• Whether the exam will expect you to also compute the permanent set ${ϵ}_C$ in strain-hardening problems, or only the new $u_r$ and ${ϵ}_y^{\prime }$.

Worked examples

Three notes cover this week at different depths:

• Lecture reconstruction — the slide-by-slide source. All six lecture examples reproduced.
• Cheatsheet — every rule, table, and recipe in one page. Includes the full quiz (mixed difficulty, reshuffles every visit).
• In-depth analysis — why each formula works, the axial-shear analogy, the geometric derivation of $G=E/[2(1+\nu )]$, and a full exam-style sample.

Common mistakes

• Single vs double shear — count cut planes on the FBD; forgetting double shear doubles your calculated $\tau$.
• Wrong shear area — a punched disk shears on its cylindrical edge $\pi dt$, not the disk face $\pi d^2/4$.
• Toughness vs resilience — the whole area vs the elastic triangle.
• Strain hardening — $E$ is unchanged; only ${\sigma }_y$ rises.
• Angle units — $\gamma$ must be in radians.
• Confusing ${\tau }_{allow}$ and ${\tau }_{fail}$ — design with ${\tau }_{allow}={\tau }_{fail}/SF$.
• Mistaking normal for shear — a vertical compression on a vertical edge is ${ϵ}_y$, not ${\gamma }_{xy}$.

Practice questions

There is no Tutorial 12 PDF. Slide 23 directs students to:

1. Mastering Physics: Shear Stress and Strain + Material Properties.
2. The workshop class — to complete Portfolio 11.
3. Tutorial class.
4. Exam preparation: notes and formula sheets.

For tutorial-style practice, revisit the Week 11 tutorial (axial stress/strain, Hooke’s law in tension) — every shear formula in Week 12 is a direct analogue of an axial formula there:

• $\sigma =N/A$ → $\tau =V/A$
• $ϵ=\delta /L$ → $\gamma \approx \delta /h$
• $\sigma =Eϵ$ → $\tau =G\gamma$

Additionally:

1. Re-do Examples 1-6 from the lecture (reproduced in lecture-summary.md) without looking at the solutions.
2. From the handbook values in the slide-20 table, recompute $G=E/[2(1+\nu )]$ for every material and check against the listed $G$.
3. Sketch a stress-strain curve and mark proportional limit, elastic limit, ${\sigma }_y$, ${\sigma }_{UTS}$, ${\sigma }_{failure}$, the elastic triangle (modulus of resilience), and the total area (toughness).
4. Re-do the disk-through-hole sizing (Example 3) with $SF=2$ on a given ${\tau }_{fail}$, not directly with ${\tau }_{allow}$.

Assessment relevance

• Exam: shear stress / strain and the stress-strain curve are near-certain question topics for CTP1.
• Portfolio 11: completed in the workshop using this lecture material.
• The $G=E/[2(1+\nu )]$ relation is a common one-line exam check.

Confidence report

• Directly supported: every formula, every worked example, every slide listed in the source files.
• Inferred: how the exam questions are likely to be framed (based on the lecture’s worked-example style).
• Gap: no Tutorial 12 PDF — practice must come from the lecture examples, Mastering Physics, and the analogous Week 11 tutorial.

Source files used

• EGD102-Physics/Lecture12_CTP1.pdf